This may be a fairly basic question as I don't have a strong background in physics. I intuitively thought that the universe must be able to be described by a Markov chain. That is, I thought you could feed the current state of the universe into a process and it would spit out the next state conditional on the laws of the universe. However, I have found no mention of the universe as a Markov chain outside of speculations on message boards.

Can the universe be described as a Markov chain or is there some reason to suggest that the next state in the universe is dependent on more than just the current state and constant universal laws?

An important issue with this image is discreteness vs. continuity: Markov chains are inherently discrete, whereas quantum mechanics (and QFT, in a specific way) has continuous time evolution.
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GerbenOct 5 '11 at 20:52

That's an excellent point. However, I was under the impression that a consequence of string theory (obviously it is just one among other hypotheses) was discrete space time--at least on a very small level. Even besides this, I was thinking that perhaps one could describe something like a continuous Markov chain as follows: x_t=f(x_{t-\delta}) and make that delta arbitrarily small to simulate continuity. Obviously, f(x_t) would have to be a continuous function itself for this to work.
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Vivek ViswanathanOct 5 '11 at 20:57

I edited your question to talk about descriptions of the universe, rather than what the universe is - for one thing, physics doesn't necessarily concern itself with what the universe "is," but also, phrasing it this way invites responses that discuss the sort of discrete approximation you mentioned, even if the universe isn't actually discrete.
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David Z♦Oct 5 '11 at 21:21

4 Answers
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The problem is that Markov chains are inherently lossy--- so in physics as it is commonly understood today, the answer is no. A Markov chain will always lose information about the initial state, as it relaxes to a stable distribution, while a quantum mechanical system does not do this. The modern understanding of a physical system is as a quantum markov chain, which is the same as a classical Markov chain with probability amplitudes taking the place of probabilities.

Excellent answer. I just have a quick question. Do we need quantum Markov chains instead of regular Markov chains because of the necessary uncertainty of the state that comes with Heisenberg's uncertainty principle?
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Vivek ViswanathanOct 6 '11 at 2:02

It's the opposite--- the uncertainty principle comes from the quantum markov-chain description of a point particle. There is no uncertainty in a quantum state--- it is a precisely defined thing. The uncertainties are in the values of position and momentum which you would get if you measure. Nobody calls it a "quantum markov chain", by the way, it's just called "quantum mechanics". But its the same thing as a Markov chain, except with amplitudes, and the transition matrix is now called the Hamiltonian.
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Ron MaimonOct 6 '11 at 5:06

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@Ron: inherently lossy? Can't a Markov chain be invertible? I understand that almost all eigenvectors vanish exponentially, but at large but finite $N$ you should still be able to find the initial condition.
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GerbenOct 6 '11 at 16:29

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@Gerben: the information is pushed into the small digits of the probability distribution, so that there is entropy production, the process is irreversible. But you are right in the abstract infinite accuracy limit--- if you know all the probability distribution to arbitrary accuracy, you never lose information.
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Ron MaimonOct 6 '11 at 16:47

First, a markov chain system must be independent of its past. However, if the state space of the possible states is expanded to include the residual history, then a system which "remembers its past" becomes a markov chain if the state space itself is large enough to include all the states with the recorded past history.

Non linear phenomena are definitely markov chains, and the universe appears to be a markov chain provided you define the state space as the exact microstate.

Why would non-linear phenomena be inherently non-Markov? Brownian motion isn't even differentiable at any point (and obviously isn't linear either), and it is Markov.
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Vivek ViswanathanOct 7 '11 at 18:59

You can always encode the past history into a bigger phase space, a random walk on histories, rather than on positions.
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Ron MaimonOct 7 '11 at 19:45

I didnt talk about differentiability at all. I used non-linear phenomena as an example where the system remembers the past history. However, it is clear that not all non-linear phenomena belongs to this category.
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armandoOct 9 '11 at 0:20